Efficient monitoring of exponential processes using memory control charts for healthcare applications

Abstract

Currently, assessing the efficiency of memory control charts both in zero-state (ZS) and steady-state (SS) contexts is a burning issue in the statistical process control (SPC). In certain cases, a memory control chart may demonstrate exceptional performance in the ZS while underperforming in the SS, and researchers assert the superiority of their proposed chart solely based on its ZS performance. Recent research indicates that a control chart receives priority over alternative control charts if it has both superior ZS and SS performances. Keeping in mind, this paper presents the design of two novel cumulative sums (CUSUM) and crosier CUSUM (CCUSUM) control charts termed and incorporating variable transformation based on the assumption of exponential distribution aimed at improving process monitoring in both ZS and SS conditions. The results of the study are derived using the Monte Carlo simulation approach. A number of different metrics are utilized using a variety of indicators. These metrics include the average and standard deviation of run length (RL) denoted by ARL and SDRL. Similarly, the performance comparison index (PCI), relative ARL (RARL), and extra quadratic loss (EQL) are also used as performance indicators. The suggested and charts are contrasted with the rival charts using these performance metrics. According to the results, the suggested control charts are more effective than other control charts when it comes to detecting shifts quickly in ZS and SS. Breast cancer (BS) and urinary tract infection (UTI) are the burning issues in the healthcare monitoring and their early monitoring is very beneficial for treatment. As a last step towards putting the suggested and control chart into practice, two examples originating from breast cancer and urinary tract infection are also included.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-47544-0.

Introduction

Variations are typically inherent in all manufacturing processes and may be categorized as either unnatural or natural. The natural variations are innocuous and are an integral part of every stable process, whereas the identification and removal of unnatural variations in the process are essential for the process stability and the fine quality of the ultimate products. The Control charts are designed to identify these unnatural variations. Control charts assist practitioners in taking timely remedial action and improving the overall quality of final goods by visually depicting process data and exposing abnormal variations. Memoryless and memory are the two broad categories of control charts. The Shewhart type charts, first developed by Shewhart¹, fall under the memoryless category and are excellent at identifying substantial changes in process parameters. However, memory-featured charts, such as those from EWMA and CUSUM, make better use of both past and current information, making them ideal for detecting small mild process changes.

Most of the earlier studies on CUSUM control charts have focused on the normal distribution. To effectively track normal variation, Acosta-Mejia et al.² suggested three distinct CUSUM control charts. Likewise, to deal with the ambiguous size of the location shift, Ryu et al.³ also presented an effective CUSUM control chart. According to them, their improved CUSUM control chart is severely affected by a difference between the actual and predicted shift sizes. In a similar vein, Tran et al.⁴ presented the CUSUM control chart for means of two populations. Additionally, Ghobadi et al.⁵ presented the fuzzy CUSUM chart to the scientific community under multivariate scenario. At the same time, Lucas and Crosier⁶, as well as Goel and Wu⁷, proposed the use of CUSUM charts head start and process location. Assuming normality, Sanusi et al.⁸ presented the CUSUM chart to effectively monitor variability of the process. To effectively monitor both the average and the variations of a process simultaneously, Ou et al.⁹ presented the adaptive CUSUM control chart. Recently, Sanusi et al.¹⁰ and Sanusi et al.¹¹ provided efficient CUSUM charts using various estimators and auxiliary-based Shewhart-CUSUM control chart. Recently, Aliet al.¹² suggested CUSUM chart for the process mean with the application of chemical engineering. Similarly, Riaz et al.¹³ provided a CUSUM based study with the ZS and SS performance criterion. Likewise, Abbas et al. ¹⁴ introduced distribution free adaptive CUSUM sign chart for mean. Also, Nazir et al.¹⁵ suggested CUSUM based study with the integration of normal and non-normal cased under RSS case. For more related recent studies see^16–18.

In practical contexts, it is frequently seen that the data produced by a process does not conform to a normal distribution. Many researchers explored different aspects of non-normal processes. For instance, Duclos and Pillet¹⁹ presented a chart that was designed to represent abnormal processes. An L estimator, which can replace the dispersion and location, which are commonly employed in Shewhart charts, is used to create this X/S-like chart. Aslam et al.²⁰ suggested a refined EWMA for the Com-Poisson distribution, showcasing enhanced efficiency in comparison to the classical EWMA chart. In a subsequent analysis, Huberts et al.²¹ examined the implications of the central limit theorem on the application of Shewhart control charts to big datasets that do not conform to a normal distribution, considering an array of distributions and varying sample sizes. In a similar way, Aslam²² advocated for the utilization of mixed EWMA-CUSUM chart, predicated on the concept of the Weibull process. In a similar vein, Aslam et al.²³ provided a control chart for the time between events (TBE) that was developed expressly for the Gamma process. Likewise, Khan et al.²⁴ suggested an efficient dispersion chart for enhanced tracking the process. Recently, Arslan et al.²⁵ provided mixed EWMA-Dual Crosier CUSUM for process location. Now, Waqas et al.²⁶ initiated enhanced EWMA for inverse Maxwell distribution with mechanical industrial applications. Similarly, an AEWMA control chart with healthcare application for inverse Maxwell process is suggested by Waqas et al.²⁷. Also, an enhanced chart for tracking the Rayleigh distribution’s scale parameter was presented by Khan et al.²⁸.

In the realm of exponential processes, where the quality feature in question adheres to an exponential distribution, researchers have suggested several improvements to memory-type control charts. Similarly, Aslam²⁹ used neutrosophic approach to create the sample plan for the exponentially data. For optimal monitoring of the exponentially distributed data, Adeoti and Malela-Majika³⁰ presented a double EWMA (DEWMA) chart. Furthermore, a novel memory chart for the exponential distribution (ED) was developed by Khan et al.³¹. It is worth noting that Aslam et al.³² introduced an EWMA chart designed to monitor ED; this chart employs probability limits to determine the average run-length (RL) (ARL) for both in-control (IC) and out-of-control (OOC) situations. Nevertheless, the studies conducted by Margavio et al.³³, Adams et al.³⁴, Abbasi et al.³⁵, and Rasheed et al.³⁶ have illuminated the limitations and drawbacks of this approach, and Rasheed et al.³⁶ delineated the appropriate methodology for the EWMA chart (denoted as ) based on the premise of exponential process (EP).

Similarly, many authors discussed the control charts with healthcare data. For example, Koetsier et al.³⁷ suggested control chart for healthcare system. Similarly, Suman and Prajapati³⁸ provided a detailed review of control charts for health sector. Likewise, Waqas et al.³⁹ highlighted the control charts with COVID data. Also, Aslam et al.⁴⁰ suggested HWMA chart for breast cancer data. For more detailed studies, see^27,36,41,42.

In contemporary discussions surrounding memory control charts, the significance of steady-state (SS) performance surpasses that of zero-state (ZS) performance⁴³. As previously noted Rasheed, et al. ⁴⁴ introduced control chart for EP, relying solely on ZS criteria.

Similarly, all the previous control charts about the exponential process are related to ZS performance evaluation criteria. In addition, if you’re looking for a substitute for EWMA control charts that can effectively monitor any process, CUSUM charts are a great option⁴⁵. Additionally, the crosier CUSUM (CCUSUM) chart was introduced by Crosier⁴⁶ as a substitute for the CUSUM mean chart. He suggested that the CCUSUM chart demonstrates greater efficiency compared to the CUSUM chart. With these considerations discussed above, the present research aims to offer two new exponential distributions based CUSUM and CCUSUM control charts (presented by and ) that outperform its predecessors, considering the quality measures of both ZS and SS. A number of different metrics are utilized in order to assess the suggested chart. The measures encompass the Average, median, and standard deviation of run length (RL) (symbolized as ARL, MDRL, and SDRL). Similarly, the relative ARL (RARL), Extra Quadratic Loss (EQL), and performance Comparison Index (PCI) are also used for overall performance measures. In addition, to underscore the real-life significance of the suggested and charts, two real-world applications from breast cancer (BC) and urinary tract infection (UTI) are presented.

The ensuing sections of this manuscript are organized as: Sections “Design structure of existing control chart” and “Proposed control charts” delineate the current methodologies as well as the proposed alternatives. Section “Assessment of performance” delineates the metrics for assessing performance. Section “ Additionally, Section “Performance comparisons” presents a comparative analysis, whereas Section “Practical application” illustrates the practical usage of the proposals. The findings and recommendations are the components that make up the final part.

Design structure of existing control chart

Suppose, the quality characteristic (QC) of interest (t), which follows an ED with a mean represented by θ, defined by the subsequent probability distribution function.

1

If T follows an ED with parameter θ, then defined as T raised to the power of , adheres to a Weibull distribution characterized by shape and scale parameters and , respectively⁴⁷. approximates a normal variate for shape parameter 3.6.⁴⁸, i.e. for .

Assume the process is beginning in a controlled condition., i.e. . In order to construct an EWMA control chart employing probability limits, the altered variable . was used by Aslam et al.⁴⁷. Later, Rasheed et al.³⁶ presented an additional transformation i.e., , which converts into standard normal variate. Rasheed et al.³⁶ proposed the following statistic

2

Here . The control limits of are by assuming

3

where L is responsible for reaching the desired IC ARL. The process is considered OOC whenever plotting statistic is less than the or greater than the ; otherwise, it is deemed to be in the IC.

Proposed control charts

This section presents the suggested charts derived from the modified variable Y. The CUSUM control chart for ED presented as chart. The statistics are presented as

4

Here, the represents the reference value, generally consists of about 50% of the shifts. Moreover, are supposed to be 0. The chart signals an OOC condition when either of the two statistics or exceeds the . Crosier⁴⁶ proposed an altered variant of the CUSUM mean chart, presented as the CCUSUM chart. The CCUSUM chart operates in a manner that is comparable to that of the CUSUM chart; hence, it is more responsive. The CCUSUM chart is equally useful for multivariate data because it may be used to keep track of more than one quality attribute at a time. The CCUSUM chart for exponential process () depends on the following statistics;

5

and

6

where = 0. The parameter represents the reference value for the chart, analogous to its function within the chart. It is very common to set to be half of the shifts when identifying the average change of size . The chart generates an OOC points under the condition that exceeds or falls below - , spoting a shift in the process, either upward or downward. The parameter delineates the rate of false alarms associated with the chart.

Assessment of performance

To assess the efficacy of the control charts, we will offer the various performance evaluation criteria here. The subsequent subsections provide additional detailed information.

Monte Carlo simulation

We use an exponential distribution with a variety of parameter choices to produce sample of , where  = 1,2,.,n. Subsequently, calculate the required statistical measures for the chart’s plotting statistic. The following is an indication of the shift in the process parameter: to , where is given in Tables A1–A4. The R software program serves as a platform for the development of an algorithm. For every single displacement of shift, a Monte Carlo simulation with 100,000 iterations carried out. The simulation process for the proposed control charts is presented in the following manner.

1. For the purpose of generating samples, the exponential distribution given in Eq. (1) is utilized.

2. The previous sample is used to calculate the transformed variable.

3. Afterwards, the and statistics should be computed by employing the use of the variable .

4. The pertinent decision intervals have been approximated.

5. Plot the associated decision intervals against the and statistics.

6. If either the or statistic fall out of the h, then note that sample number because it is a RL.

7. Do steps 1 to 5 100,000 times, keeping an eye on the RLs on every occasion.

8. Find the mean of 100,000 different RLs to find the value.

9. Ultimately, ARLs for different number of samples and parameter combinations are determined.

10. Repeat steps 1 through 9 after gathering additional samples under shift from the corresponding distribution to ascertain the values.

ARL measure

Among the many metrics to assess the performance of a control chart over a shift, the ARL indicator is among the most popular and well-known. There are two types of symbolized named as and . The number of false alarms that could occur when a process was in an IC state was reduced with an appropriately sized . Still, the needs to be very little to notice the change rapidly. For a control chart to be more effective, it must have a lower at the same level.

Indicators for evaluating overall performance

The EQL, RARL, and PCI indicators of performance can be used to determine the total effectiveness of a control chart. Additional details are provided here. The RARL is widely employed for evaluating OP of a control chart. The RARL is documented as.

  

Additionally, the ARLs of the competitor and BM charts are represented by the ARL(δ) and respectively. At a particular , a chart is considered to be a BM if it has a lower ARL than other control charts⁴⁹. At this point, it is assumed that the RARL value included within the BM chart is equal to one. If the RARL value is larger than 1, the BM chart is thought to be better than the competing chart.

The EQL is calculated by averaging the ARL across all shifts using squared as the weighting factor⁵⁰. Mathematically, it is written as.

where and represent the maximum and minimum deviations of the range, respectively. As far as control charts are concerned, the one with the lowest EQL is clearly the best.

The PCI is another indicator used here and its definition is.

In the event that the PCI is bigger than 1, the BM chart outperforms the competing control chart⁵¹.

IC behaviour of and charts

The parameters and serve as the foundation for IC behaviour of the and charts. To conduct the simulation study for the fixed sample size 1, the values of are established as 0.10, 0.25, 0.50, 0.75, 1.0, 1.5, 2.0, and 2.5 (see⁵² and⁵³). Subsequently, each value is utilized to calculate the control limit , ensuring that the target approximates 200 and 370. Tables A1–A4 and A8–A11 display the RL matrices of the and charts. The following are the primary conclusions drawn from the suggested the and charts:

• The RL distributions of the and charts exhibit a positive skewness (refer to Tables A1 and A3).

• In the context of the proposed and charts, it is noted that the SDRL values are less than the ARL, as highlighted in Tables A1–A4.

• In order to get IC ARL 200 and 370, the value reduces as increases. For example, value decreases from 7.974, 5.79 and 3.75, the values of increases from 0.10 to 0.25, 0.50 respectively, in the chart with=1 and .

OOC behaviour of and charts

To assess the OOC behavior of the charts with sample size 1 and an ARL of 200 and 370, various combinations of design parameters will be examined. We used a range of values, including 1.0, 1.05, 1.10, 1.15, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.5, 3.0, 4.0, 5.0, and 6.0, to examine the OOC behavior. Tables A1–A4 and A8–A11, respectively, show the RL behaviour under OOC criteria for the suggested control chart. Furthermore, the comprehensive performance metrics for the proposed versus existing competitors’ charts have been figured and are presented in Table A15. The findings presented in Tables A1–A4 and A8–A11 yield a series of conclusions articulated as follows.

• The proposed control charts demonstrate enhanced efficacy for lower values of k regarding the RL characteristics (refer to Figs. 1 and 2).

• The of and charts decline with the rise in the k values. As an illustration, when rises by 0.01 to 2.00, the control chart reaches the values of 186.81, 200.49, 217.60, 228.54, 233.46, 245.38, 249.23, respectively, in the SS condition for  = 0.10,  = 370 (see Table A11).

• Table A15 specifies the control chart overall best in term of performance when approaches the smallest value i.e., 0.10.

Fig. 1.

chart for various using .

Fig. 2.

chart for various using .

Performance comparisons

The ZS and SS comparisons between the proposed and existing charts are provided in this section. The comparisons are performed at 200, 370. The subsequent subsections contain more information on the contrasting results.

Zero state performance comparison with and charts

The comparison of ZS is based on the assumption that the shift is happening at the beginning of the process. Tables A1–A7 display the ZS RL outcomes for specific shifts linked to , , , and control charts for  = 200, 370. Table A15 presents the measures that are employed to appraise the OP under the assumption that ZS is present.

When equated to the existing and control charts, the suggested and charts are effective at tracking moderate and minor shifts. For example, at shifts 1.05,1.10, 1.15 and  = 200 the and provide the values (133.69, 98.33, 75.00) and (115.27, 79.00,59.05), whereas the existing control chart ( has the value equal to (171.68, 127.09, 95.89) and chart has values (181.56, 149.93, 122.84) (see Table A1–A2 vs. Table A5–A7). Figures 3, 4, 5 and 6 also show the relative advantages of suggested and charts over and control charts practically.

Fig. 3.

and charts under = 370.

Fig. 4.

and charts under .

Fig. 5.

and charts under = 370.

Fig. 6.

and charts under = 370.

Moreover, the proposed and control charts keep their priority by keeping the smaller SDRL than the and . For instance, with shifts 1.05, 1.15, and = 370, the proposed and control charts at the common  = 0.1 provide the values () and (191.80, 77.05), whereas the control chart ( has the value (308.14, 146.24) and chart has values () (see Tables A2, A4, A6, A7).

The proposed and charts outperform the current and charts, according to the OP metrics under ZS. For instance, chart and has , , and values are 107.84, 1.23, 1.29 and 87.33,1.00,1.00, whereas the and control charts have 118.21, 1.35, 1.29 and 171.31, 1.96, 2.09, respectively (see Table A15). According to the ZS criteria, the chart outperforms the others when comparing overall performance.

Steady state performance comparison with and charts

The SS performance assessment criterion is restored than the ZS criterion, as stated by Knoth et al.⁵⁴. For the sake of SS, it is believed that the shift does not take place at the initial step of the process but rather at a future stage in the process. This subsection compares , , , and control charts using SS. Tables A8–A14 exhibit the RL outcomes of the control charts under the assumption of SS. The outcomes of the comparisons of overall performance in the SS area are detailed in Table A15.

Under the assumption of SS, the proposed and charts Achieve enhanced performance compared to the existing and charts to track tiny to medium shifts. Let us suppose, at and the proposed and control charts for provide the values (159.92, 138.94) and (149.81, 128.00), whereas the control chart ( has thevalues (179.98, 155.40) and chart has values (192.65, 175.22) (see Table A8, A10 vs. Tables A12, A14). Similarly, the dominance of the suggested and charts against the and charts can also be observed through additional values within the tables. The SDRL performance is as essential when compared to the SS regardless of the presence of the ARL. The suggested and control charts are more advantageous than the and under criteria. As an example, with shifts 1.05, 1.10 and  = 370, the and control charts with provide the values (197.92, 130.46) and (201.73, 129.13), whereas the control chart ( has the values (306.97, 241.83) and chart has values (301.28, 251.12) (see Tables A9, A11 vs. Tables A13, A14).

Moreover, in SS, the and charts exhibit surpassed OP than the and charts. As an illustration, with the values of the and charts with are 291.53, 1.05, 1.05 and 277.25, 1.00, 1.00, suggesting a better overall performance against the and charts yield the values of 296.87, 1.04, 1.05 and 336.02, 1.17, 1.19, respectively (see Table A15). However, the chart is best than all others competitors for overall performance comparison criterion under SS.

Performance comparison under standard error of ARL

The standard error (SE) of ARL (SE_ARL) estimate is another criterion for assessing the performance of proposed with existing control charts. The SE_ARL of the , , and charts are presented in Table A1–A7. For small shift, the suggested and are more sensitive than the existing and charts. For example, at ARL = 200, SE_ARL of and charts decrease from 5.797 to 0.449 and 7.5 to 0.407 for the shifts 1–1.5 respectively. Similarly, SE_ARL of and charts decrease from 6.966 to 0.653 and 6.337 to 1.485, respectively. Likewise, at shift 2.5, the , , and charts provide SE_ARL values 0.142, 0.131, 0.167, and 0.291, respectively. For extremely large shifts like 4–6, the SE_ARL values are 0.05, 0.473, 0.50, 0.61 for the , , and charts respectively. Hence, the proposed and are more sensitive than and charts for the monitoring of small to large shifts in the exponential process. For extremely large shifts, chart is also performing good.

Practical application

In order to demonstrate the potential real-world applications of the suggested and charts, this section offers two case studies. Analyses of breast cancer survival rates are presented in Section “Application with BC data”, while a data set pertaining to UTIs is used in Section “Application with urinary tract infection data” for practical purposes. complications. The proposed control charts are highly beneficial for keeping an eye on patients in a clinical context since they can notice changes in a patient’s health early on. For example, these charts let doctors look at patient health data in BC survival monitoring to discover early signals of therapy failure or negative occurrences. For UTIs, they can keep track of how many people get sick over time and let people know early about possible outbreaks or treatment resistance. This is vital for both taking care of patients and responding to public health issues. Adding these charts to their clinical systems lets healthcare providers see real-time updates on their patients’ health. Patients obtain the proper treatment at the right time and the right modifications are made, which leads to improved outcomes. The proposed charts are meant to help with early action in healthcare by spotting minor changes in important health indicators like infection rates or patient survival data. This gives healthcare workers a chance to respond before a patient’s condition gets worse, which could minimize the risk of mortality and the overall expense of the sickness. For figuring out how long people with cancer will live, noticing tendencies that are getting worse early on might lead to swift changes in treatment plans that will help patients get well. In the same way, learning about increased incidence of urinary tract infections early on can lead to earlier isolation measures, which will halt infections from spreading in hospitals and improve the total level of care.

Application with BC data

BC is the utmost harmful and fastest spreading non-metastatic illness among women around the world. If it is caught early, 70–80% of the time it can be treated successfully²³. A serious global threat has been brought to light by the fact that around 17 lac new instances of BC have been reported per year over the course of the past five years⁵⁵. Irrespective of the nation’s economic status, breast cancer imposes challenges in early detection, prevention, and treatment. Poor nations still don’t have what it takes for early detection, effective treatment, and preventative screening, which explains why their death toll is climbing despite falling incidence rates. Because it is a vicious and quickly kills millions of people per year, we must do all in our power to find ways to prevent, detect, and treat the disease quickly so that we can lessen its impact and increase survival rates.

This particular example evaluates the dataset that was compiled by Awodutire et al.⁵⁶, which displays the length of time that the patients of this epidemic had survived. Ladoke Akinotola University in Nigeria was the location where breast cancer patients were treated between the years 2000 and 2014. This information was collected from those patients. In addition, Awodutire et al.⁵⁶ found that these data are best represented by the exponential distribution.

The and tools are applied in combination with the existing control chart to analyze this data. Providing a way to track the death rate of breast tumor patients is the main goal of this program.

This will facilitate the development of practical preventive measures, bolster the establishment of campaigns for the promotion of public health, lessen the impact of the illness and improve chances of survival. The analysis reveals a satisfactory alignment, as evidenced by a KS statistic of and a , indicating that these data are consistent to ED. The fitness can also be seen in Q-Q plot and empirical versus fitted CDF graph in Figs. 7 and 8.

Fig. 7.

Q-Q plot of BC data.

Fig. 8.

Empirical verses fitted CDF plot of BC data.

To execute the proposed and existing control charts, the IC is set at 104.67 as an mean of the sample by following Anwar et al.⁵³. The initial 30 samples are designated as IC, while a little shift 1.16 is applied to the remaining samples, as documented in existing studies^53,57. The parameters arrangement of various control charts are assumed as under: For the ( = 0.25,  = 5.79), for ( =  0.1,  =  2.694), and ( = 0.25,  = 7.68) with prespecified .

The data presented in Figs. 11, 12 and 13 reveal that the proposed , , charts display the first OOC indication at 47, 45, and 49, respectively. In a similar way the reveals merely two OOC points, whereas the proposed and control charts detect four and five OOC points, respectively. The findings suggest the suggested charts demonstrate higher sensitivity in identifying minor variations in process when contrasted with the control chart.

Fig. 11.

The implementation of the chart for cancer data when .

Fig. 12.

The implementation of the chart for cancer data when .

Fig. 13.

The implementation of the chart using cancer data for .

Application with urinary tract infection data

This segment offers a practical implementation of the and charts employing the UTI sample (quantified in ng/ml). Khan et al.³¹ utilized this real-life sample for the montoring the EP in past. The findings suggest that the number of UTIs that occur in male patients after they leave the hospital is on the rise, with an average duration of 0.21 days. It is possible for the symptoms of a UTI to differ depending on a number of factors, such as age, gender, and financial situation, in addition to unique risk factors for UTIs that are only found in women. In comparison to their male counterparts, the urethras of females exhibit a distinct reduction in size. This directly minimizes the distance that pathogens need to traverse to reach the bladder. Engagement in sexual activity has been associated with a heightened occurrence of urinary tract infections. The utilization of diaphragms as a contraceptive method could potentially increase the likelihood of urinary tract infections. Moreover, the potential for risk is markedly heightened in the presence of an unexpected companion of sexual activity. Mostly, the medical centers seeks to systematically track the occurrence of patients who are released following the onset of a urinary tract infection during their hospitalization. This initiative is designed to swiftly detect any rises in infection rates and to evaluate the effects of upcoming procedural or material changes on the mitigation of infections. Table A13 presents the data pertaining to UTIs. The KS test is performed for evaluating the adequacy of data implementation. The KS = 8.259 (p-value = 0.25) show that the outcomes align with an study distribution, indicating a satisfactory fit (also see Q-Q plot and empirical versus fitted CDF graph in Figs. 9 and 10).

Fig. 9.

Q-Q plot of UTI data.

Fig. 10.

Empirical versus fitted CDF of UTI data.

In order to account for OOC performance, a shift of 1.22 was applied across the last fifteen samples, with the first fifteen serving as an IC. In this case, the chart is designed for k = 0.50, h = 3.75, L = 2.825, λ = 0.2 for chart, and k = 0.50, h = 4.685 for chart at prespecified .

The initial OOC spots are found at 18th sample both for and charts, but at the 25th sample on the chart (Figs. 14, 15 and 16). Furthermore, the suggested , and charts validate eleven, twelve, and two OOC signals, respectively. This illustrates that the and charts exhibit higher performance compared to existing chart.

Fig. 14.

The implementation of the chart for UTI data.

Fig. 15.

The implementation of the chart using UTI data.

Fig. 16.

The implementation of the chart for UTI data.

Final remarks and suggestions

In order to track the EP in both ZS and SS conditions, this research suggests using two control charts: CUSUM and crosier CUSUM (CCUSUM). The suggested charts are presented as and to spot the shifts in the EP efficiently and effectively. Numerical outcomes are obtained using the Monte Carlo simulations. Famous performance measures like ARL, MDRL, and SDRL are employed to track the individual shifts in the process. Similarly, for the monitoring of overall performance, different performance indicators like EQL, PCI, and RARL are obtained. The aforementioned performance metrics are used to compare the suggested charts with well-known exponential-based charts such as and . Based on the contrasts, it appears that the suggested and charts outperform the rival charts regarding SS and ZS conditions. Similarly, both suggested control chart’s SS and ZS ARL and SDRL are contrasted. The and charts have minimum ARL and SDRL values under ZS as compared to SS. Similarly, the chart has higher quality outcomes compared to the chart under ZS and SS criterions. Lastly, the suggested charts are put into practice using two true data sets regarding the health care sector. Furthermore, the results of true data indicate the suggested charts are superior to the existing rivals. This research can be additionally prolonged to include other abnormal distributions like Weibull and Gamma and multivariate scenarios.

Supplementary Information

Below is the link to the electronic supplementary material.

List of symbols

ZS

Zero-state

SPC

Statistical process control

CCUSUM

Crosier CUSUM

EP

Exponential process

CCUSUM for EP

CUSUM for EP

PCI

Performance comparison index

RARL

Relative ARL

BC

Breast cancer

AEWMA

Adaptive EWMA

OOC

Out-of-control

QC

Quality characteristic

Improved EWMA for EP

Transformed variable

Lower control limit of   

Upper

parameter

statistic

OOC ARL

BM

Benchmark

ARL(δ)

ARL at

Maximum

KS

Kolmogorov-Smirnov

SS

Steady-state

CUSUM

Cumulative sums

ED

Exponential distribution

RL

Run length

ARL

Average RL

SDRL

Standard deviation RL

MDRL

Median RL

EQL

Extra quadratic loss

UTI

Urinary tract infection

DEWMA

Double EWMA

IC

In-control

parameter

statistic

L

limit constant

Upper control limit of

Lower

Decision interval

Shift

OP

Overall performance

IC ARL

ARL of BM at

Minimum

CDF

Cumulative distribution function

Author contributions

S. M. A., N. A., and C-Y. L. participated in composing the initial text, executing mathematical investigations, and carrying out computer-based results. N. A. authenticated all findings, reorganized the text, and oversaw financial procurement. All authors examined and sanctioned the last draft of the article.

Funding

The contributors would like to express their gratitude to King Khalid University's Deanship of Research and Graduate Studies for providing financial support for this project through the Large Research Project (RGP2/50/46), and to National Chung Hsing University for financial support through the National Science and Technology Council (Taiwan) project (Project No. NSTC 113-2118-M-005-004-MY3).

Data availability

Upon appropriate request, the datasets utilized and/or evaluated for this investigation could be acquired by contacting the relevant authors.

Declarations

Competing interests

The authors declare no competing interests.